According to the dictionary Webster’s 1913, which can be accessed through \htmladdnormallinkHyperDictionary.comhttp://www.hyperdictionary.com/, mathematical meaning of the word operationMathworldPlanetmath is: “some transformationPlanetmathPlanetmath to be made upon quantities”. Thus, operation is similarPlanetmathPlanetmath to mapping or function. The most general mathematical definition of operation can be made as follows:

Definition 1

Operation # defined on the sets X1,X2,,Xn with values in X is a mapping from Cartesian product X1×X2××Xn to X, i.e.


Result of operation is usually denoted by one of the following notation:

  • x1#x2##xn

  • #(x1,,xn)

  • (x1,,xn)#

The following examples show varietyMathworldPlanetmath of the concept operation used in mathematics.


  1. 1.

    Arithmetic operations: additionPlanetmathPlanetmath (http://planetmath.org/Addition), subtractionPlanetmathPlanetmath, multiplication (http://planetmath.org/Multiplication), division. Their generalizationPlanetmathPlanetmath leads to the so-called binary operationsMathworldPlanetmath, which is a basic concept for such algebraic structuresPlanetmathPlanetmath as groups and rings.

  2. 2.

    Operations on vectors in the plane (2).

  3. 3.

    Operations on vectors in the space (3).

  4. 4.

    Some operations on functions.

    • Composition.

    • Function inversePlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath.

In the case when some of the sets Xi are equal to the values set X, it is usually said that operation is defined just on X. For such operations, it could be interesting to consider their action on some subset UX. In particular, if operation on elements from U always gives an element from U, it is said that U is closed under this operation. Formally it is expressed in the following definition.

Definition 2

Let operation #:X1×X2××XnX is defined on X, i.e. there exists k1 and indexes 1j1<j2<<jkn such that Xj1=Xj2==Xjk=X. For simplicity, let us assume that ji=i. A subset UX is said to be closed under operation # if for all u1,u2,,uk from U and for all xjXjj>k holds:


The next examples illustrates this definition.


  1. 1.

    Vector space V over a field K is a set, on which the following two operations are defined:

    • multiplication by a scalar:

    • addition


    Of course these operations need to satisfy some properties (for details see the entry vector space). A subset WV, which is closed under these operations, is called vector subspace.

  2. 2.

    Consider collectionMathworldPlanetmath of all subsets of the real numbers , which we denote by 2. On this collection, binary operation intersection of sets is defined:


    Collection of sets 2:


    is closed under this operation.

Title operation
Canonical name Operation
Date of creation 2013-03-22 14:57:23
Last modified on 2013-03-22 14:57:23
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 10
Author rspuzio (6075)
Entry type Definition
Classification msc 03E20
Related topic Function
Related topic Mapping
Related topic Transformation
Related topic BinaryOperation
Defines closed under
Defines arithmetic operation